hw6 - determine the most ecient way to combine these legs...

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AMS/MBA 546 (Fall, 2010) Estie Arkin Network Flows Homework Set # 6 Due Tuesday, October 26, 2010. Suggested reading Chapter 6 of the text [AMO]. 1. Suppose that there are n men, m women and k marriage brokers. Each broker has a list of some of the men and women as clients and can arrange marriages between any pairs of men and women on that list. (Note that each person can be on lists of several brokers.) For this problem, all marriages must be heterosexual and all people are monogomus. (a). Translate the problem of ±nding the most marriages to that of ±nding a max ²ow. Make sure to clearly describe your graph. If you draw the graph, make sure to indicate the source and terminal and all capacities. (b). Assume an additional restriction, that each broker i can arrange at most a i marriages. Modify your formulation from part (a) to ±nd the most marriages. 2. [6.32] An airline has p ²ight legs that it wishes to service by the fewest possible planes. To do so, it must
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Unformatted text preview: determine the most ecient way to combine these legs into ight schedules. The starting time for ight i is a i and the nishing time is b i . The plane requires r ij hours to return from the destination of ight leg i to the start (origin) point of ight leg j . Suggest a way to solve this problem using ow. 3. [6.40] Given a directed graph G = ( N, A ) with upper bounds on the ow u ij which are integer, and nodes s, t N , we know that there is a max ow from s to t that is integer. Suppose that instead, a helpful friend gives us a max ow x ij where some of the ows are noninteger. Show how to convert this ow into an integer maximum ow. The running time of your algorithm should be O ( mf ) wher m is the number of arcs and f is the number of arcs with fractional (noninteger) ow on them. Hint: Send ows along cycles....
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